KS3 Maths Now Textbook

351 25.4 12 Show that the expression in the top brick of this algebra wall can be written as 5(x + 3). 13 Show that the expression in the top brick of this algebra wall can be written as 6(x – 1). 14 Show that the given expressions are equivalent expressions for the perimeters of the shapes in question 5. a 3(t + 2) b 4(k + 2) c 4(c + 4) d 3(a + 8) e 5(x + 2) 15 This shape has been divided into rectangles in two different ways. x – 3 x – 3 9 x 3 3 7 2 2 a Use the first diagram to write an expression for the area of the whole shape. b Use the second diagram to write a different expression for the area of the whole shape. c Show that the expressions are equivalent. Solve problems Work out the expressions missing from the blank bricks. Working backwards: In the bottom row, the middle brick is 2x + 5 – (x + 3) = 2x + 5 – x – 3 = x + 2 In the middle row, the right brick is 4x + 11 – (2x + 5) = 4x + 11 – 2x – 5 = 2x + 6 In the bottom row, the right brick is 2x + 6 – (x + 2) = 2x + 6 – x – 2 = x + 4 x + 3 2x + 5 4x + 11 x + 3 x + 2 2x + 5 2x + 6 4x + 11 x + 4 16 Work out the expressions missing from the blank bricks. x + 1 2x + 8 4x + 10 x + 5 x + 4 2x + 2 x –1 2x –4 x – 3 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 351 2867_P001_680_Book_MNPB.indb 351 24/03/2020 15:33 4/03/2020 15:33

352 Chapter 25 17 Work out the expressions missing from the blank bricks. 2x – 1 x + 1 6x x + 3 18 For each question, mark any necessary lengths on your diagram. a Sketch a rectangle with area abc. b Sketch a rectangle with perimeter 6x + 4y. c Sketch a triangle with area 8x. d Sketch a rectangle with area 5x + 10. 25.5 Using index notation ● I can write algebraic expressions involving powers You have already used some powers such as x × x = x2 and y × y × y = y3 . You can have powers larger than 3. a × a × a × a = a4 This is a to the power 4. The number 4 is called the index. Develop fluency Write each expression in index form. a 5 × 5 × 5 × 5 b m × m × m c t × t 2 d n × 2n × 4n e x × 2y × x a 5 × 5 × 5 × 5 = 54 b m × m × m = m3 c t × t 2 = t × t × t Because t 2 = t × t. = t 3 d n × 2n × 4n = 2 × 4 × n × n × n You can change the order. Put the numbers together. = 8n3 2 × 4 = 8 and n × n × n = n3. e x × 2y × x = 2 × x × x × y Put the number in front. = 2x2y Leave out the × signs. 1 Write each expression in index form. a 4 × 4 × 4 b 3 × 3 × 3 × 3 × 3 × 3 c 10 × 10 × 10 × 10 × 10 2 Write each of these out in full. a 24 b 53 c 65 d 96 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 352 2867_P001_680_Book_MNPB.indb 352 24/03/2020 15:33 4/03/2020 15:33

353 25.5 3 Write these numbers as a power of the base 2. The first one has been done for you. a 8 = 23 b 16 c 32 d 64 4 Write each expression in index form. a a × a × a × a b r × r × r c b × b × b × b × b d m × m × m × m × m × m e 4a × 3a f p × 2p g 2g × 3g × 2g h k × 4 × 2k × k × 3k 5 Write each expression as simply as possible. a f + f + f + f + f b w × w × w × w c c + c + c + c + c + c + c d k × k × k × k e D + D + D + D + D + D 6 Write each expression as simply as possible. a a × b × a b x × y × y × 6 c t × u × t × u d d × 2d × c e a × 2b × b f w × x × 2w g 2t × 2t × u h 21e × 3f × 4e 7 Write each expression as simply as possible. a c 2 × c b 5c × 3c4 c c3 × c 3 d 9c3 × 7c3 8 Write each expression as simply as possible. a t × t 3 b 2t × 3t 3 c t 2 × t 2 d 5t 2 × 6t 2 9 Expand the brackets. Write your answer as simply as possible. a 4w(5w − w2 ) b 7x2 (x + 2x) c 8yz(3y2 – 5z) 10 Write each expression as simply as possible. a 2ab3 × 5ab5 b (4c2 )3 c (2d2 e)4 Reason mathematically Jo says that 5j and j 5 are the same. Is she correct? Give a reason for your answer. She is not correct because 5j = j + j + j + j + j but j 5 = j × j × j × j × j. 11 Explain the difference between 6w and w 6 . 12 The formula for the area of a triangle is A = 1 2 b × h where b is the base and h is the height. The area of this triangle is 1 4 b2 cm2 . What does this tell you about the height? 13 Show that 26 has the same value as 43 . b cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 353 2867_P001_680_Book_MNPB.indb 353 24/03/2020 15:33 4/03/2020 15:33

354 Chapter 25 14 Fill in the missing term. 2a4 × 3a2 × 4a × … = 48a9 Solve problems Work out an expression for the volume of this cube. Volume is 2x × 2x × 2x = 8x3 cm3 2x cm 15 Work out an expression for the volume of each cuboid. a x cm x cm 4 cm b y cm y cm y cm c t cm t cm 2t cm d k cm 2k cm 2k cm e n cm 2n cm 3n cm 16 You are given that 210 = 1024. Work out the value of a 29 b 211 17 Joan says that to simplify x a × x b , you just multiply the powers so the answer is x ab. Give an example, using numbers, to show that she is not correct. Now I can… simplify algebraic expressions involving the four basic operations manipulate algebraic expressions write algebraic expressions involving powers simplify algebraic expressions by combining like terms identify equivalent expressions remove brackets from an expression © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 354 2867_P001_680_Book_MNPB.indb 354 24/03/2020 15:33 4/03/2020 15:33

355 26 Working with fractions 26.1 Adding and subtracting fractions ● I can add or subtract any two mixed numbers Before you can add or subtract two fractions you need to make sure they are written with the same number in the denominator. Remember to give an answer in its simplest terms. Develop fluency Work these out. a 4 1 2 + 2 2 3 b 3 1 4 – 7 12 a 4 1 2 + 2 2 3 = 9 2 + 8 3 Write the mixed numbers as improper fractions. = 27 6 + 16 6 6 is a multiple of 2 and 3 so change both fractions to sixths. = 43 6 Add the numerators. The denominator stays the same. = 7 1 6 43 ÷ 6 = 7 remainder 1 b 3 1 4 − 7 12 = 13 4 − 7 12 39 12 − 7 12 = 32 12 Change 13 4 into twelfths and then subtract. = 8 3 Divide numerator and denominator by 4 to find an equivalent fraction. = 22 3 1 Work these out. a 3 4 + 1 8 b 3 4 + 1 6 c 3 4 − 2 3 d 3 4 − 7 10 2 Work these out. a 1 1 2 + 2 3 4 b 7 8 + 2 1 4 c 3 3 4 + 1 5 6 d 6 1 4 + 2 2 3 3 Work these out. a 4 1 2 − 2 3 4 b 2 7 8 − 2 1 4 c 4 1 4 − 1 5 8 d 6 3 4 − 6 2 3 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 355 2867_P001_680_Book_MNPB.indb 355 24/03/2020 15:33 4/03/2020 15:33

356 Chapter 26 4 Work out a the sum b the difference of these two fractions. 7 12 3 8 5 Work these out. a 1 1 2 + 2 3 4 + 3 5 8 b 2 3 4 + 1 1 2 + 3 1 3 c 2 2 3 + 1 1 6 + 3 5 9 6 Work out the missing numbers in these calculations. a 4 1 3 + … = 8 b 2 3 10 + … = 5 9 10 c 1 2 3 + … = 4 1 2 7 Work these out. a 1 1 2 + 2 3 4 − 1 1 8 b 4 3 5 − 2 1 2 − 1 1 10 c 2 1 2 + 1 1 4 + 5 8 8 Work out the missing fraction in each of these calculations. a 7 8 + 4 5 + ? = 2 b 2 3 5 + 3 1 4 + ? = 6 c 1 5 6 + ? – 2 1 3 = 3 7 8 9 x = 3 3 4 and y = 2 2 5 Work these out. a x + y b x − y c 2x + y d 2x – y Reason mathematically a Work out the lengths marked a and b in this shape. b Work out the perimeter of the shape a a = 3 1 4 − 2 3 8 = 7 8 b =2 1 2 − 5 8 = 1 7 8 b The perimeter is the distance around the edge of the shape. 7 8 + 1 7 8 + 2 3 8 + 5 8 + 3 1 4 + 2 1 2 = 11 1 2 a b 2 1 2 3 1 4 2 3 8 5 8 10 Work out the perimeter of this triangle. 11 The perimeter of this triangle is 13 1 2. What is the length of the base of the triangle? 12 a Work out the difference between the length and the width of this rectangle. b Work out the perimeter of the rectangle. 13 A quadrilateral has a perimeter of 12 3 4. The lengths of three of the sides are 4 1 2, 2 1 8 and 3 1 4. What is the length of the fourth side? 4 42 3 33 4 43 4 32 5 81 3 41 2 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 356 2867_P001_680_Book_MNPB.indb 356 24/03/2020 15:33 4/03/2020 15:33

357 26.1 Solve problems Look at these two equations. 4 5 16 + x = 7 3 8 x − y = 2 1 3 Work out the value of y. In the first equation, x = 7 3 8 − 4 5 16 = 59 8 − 69 16 = 118 16 − 69 16 = 49 16 = 3 1 16 Using x = 3 1 16 in the second equation, 3 1 16 − y = 2 1 3. So, y = 3 1 16 − 2 1 3 = 49 16 − 7 3 = 147 48 − 112 48 = 35 48 14 Here are four numbers. 2 2 3 2 1 6 2 1 4 2 5 12 Which two fractions have a a sum of 5 1 2? b difference of 1 12 ? 15 Look at this sum. 3 5 6 + 1 2 9 = 5 1 18 Use the sum to write down the answer to each of these calculations. a 5 1 18 − 1 2 9 b 4 5 6 + 2 2 9 c 1 5 6 + 6 2 9 d 5 1 18 − 3 2 9 16 A plumber has 1 5 6 m of pipe. He needs to cut a piece measuring 7 8 m from it. How much pipe is left over? 17 A jug has 1 3 4 litres of lemonade in it. Semma pours 5 8 litres into one glass and 1 3 litre into another glass. How much lemonade is left in the jug? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 357 2867_P001_680_Book_MNPB.indb 357 24/03/2020 15:33 4/03/2020 15:33

358 Chapter 26 26.2 Multiplying fractions ● I can multiply two fractions When you are multiplying fractions, the order of the multiplication does not matter. This is the same as when you multiply integers. For example, 2 × 3 = 6 and 3 × 2 = 6. Similarly, 2 3 × 2 5 = 4 15 and 2 5 × 2 3 = 4 15 . To multiply two fractions, you multiply the numerators and multiply the denominators. There is no need to find a common denominator. To multiply a fraction by an integer, you multiply the numerator by the integer and leave the denominator unchanged. For example, 200 × 3 4 = 600 4 = 150. Often the word ‘of’ is used instead of ×. So, 2 3 of 2 5 and 2 3 × 2 5 mean the same. Remember always to give an answer in its lowest terms. Develop fluency Work these out. a 3 4 of 1 2 b 2 3 × 3 5 a 3 4 of 1 2 = 3 4 × 1 2 = 3 1 4 2 = 3 8 b 2 3 × 3 5 = 6 15 2 × 3 = 6 and 3 × 5 = 15 = 2 5 Simplify the fraction as much as possible. 1 Work these out. a 2 × 3 8 b 3 × 3 4 c 3 × 4 5 d 4 × 3 8 e 1 5 of 3 f 4 5 of 3 g 2 3 of 4 h 5 6 of 2 2 Work these out. a 1 2 × 1 3 b 1 2 × 3 5 c 1 3 × 1 4 d 1 3 × 2 3 3 Work out 2 3 of: a 1 2 b 3 4 c 2 3 d 4 5 e 1 8 f 5 6 The numerator is 3 × 1 and the denominator is 4 × 2. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 358 2867_P001_680_Book_MNPB.indb 358 24/03/2020 15:33 4/03/2020 15:33

359 26.2 4 Work these out. a 2 3 × 1 4 b 3 5 × 3 4 c 4 9 × 3 8 d 4 5 × 5 12 5 Work these out. a 1 2 × 1 2 b 2 3 × 2 3 c 3 5 × 3 5 d 3 4 × 3 4 6 Work these out. a 5 8 × 4 5 b 1 10 × 5 6 c 7 8 × 2 3 d 5 6 × 3 4 7 Work these out. a ( 1 4) 2 b ( 3 4) 2 c ( 2 3) 2 d ( 4 5) 2 8 Work these out. a ( 1 2 + 1 4) × 3 5 b ( 2 3 + 1 6) × 3 10 c ( 1 2 − 1 6) × 3 4 9 s = 5 8 and t = 2 3 Work these out. a st b ts c s 2 d t 2 Reason mathematically Zafar has raised £2400 for some charities. He is going to give 3 8 of the money to charity A, 7 16 to charity B and the remainder of the money to charity C. How much money does each ch arity receive? Charity A: 3 8 of £2400 = 3 8 × 2400 = 3 8 × 2400 1 = 7200 8 = £900 Charity B: 7 16 of £2400 = 7 16 × 2400 = 7 16 × 2400 1 = 16800 16 = £1050 Charity C: £2400 – £900 – £1050 = £450 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 359 2867_P001_680_Book_MNPB.indb 359 24/03/2020 15:33 4/03/2020 15:33

360 Chapter 26 10 a Belinda uses 1 3 of a tin of polish every time she cleans her car. How many tins has she used after cleaning the car 12 times? b Trevor spends one quarter of an hour every day tidying his room. How many hours does he spend tidying his room in four weeks? 11 Here is a multiplication. 5 12 × 4 15 = 1 9 Use this result to work these out. a 5 12 × 8 15 b 5 12 × 2 15 c 5 6 × 4 15 12 4 5 × 3 4 × 2 3 × 1 2 = 24 120 = 1 5 5 6 × 4 5 × 3 4 × 2 3 × 1 2 = 120 720 = 1 6 a Show that if you extend the series by placing 6 7 × at the front of the calculation, the answer will be 1 7 . b Without doing the calculation, what is the answer to the following: 9 10 × 8 9 × 7 8 × 6 7 × 5 6 × 4 5 × 3 4 × 2 3 × 1 2 13 Tom has an amount of money that he wants to share among his grandchildren. He gives half of his money to his daughter, Jenny, and the o ther half to his son, Mike. He asks Jenny and Mike to share the money he has given them evenly among their children. Jenny has three children and Mike has two children. Tom assumes that each of his grandchildren will be given 1 5 of the original amount. Explain why Tom is wrong to assume this. Solve problems An area of land is being split up into allotments. The land is 1 4 km wide and 2 5 km long. Each allotment is 1 400 of the total area size. How many square metres is each allotment? The size of the land is 1 4 × 2 5 = 2 20 = 1 10 km2. 1 km2 = 1000 m × 1000 m = 1 000 000 m2 Each allotment will be 1 400 × 100 000 = 250 m2 14 1 2 × a b = 3 8 Work out the values of a and b. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 360 2867_P001_680_Book_MNPB.indb 360 24/03/2020 15:33 4/03/2020 15:33

361 26.3 15 A poster uses red, blue and yellow inks. Of the ink used, 2 9 is red and 1 6 is blue. a To print the poster, 36 ml of ink is used. Calculate the amount of each ink used. b Another poster uses 6 ml of red ink. How much ink is used altogether? 16 Jamie and Ollie are plastering three walls of a rectangular living room. The area is 30 m2 in total. a Jamie does 3 5 of the total and Ollie does the rest. How many square metres does each person do? b The boss pays them a total of £90 divided in the ratio of the amount they complete. How much does each person get? c The area of the fourth wall in the room is 5 m2 . What fraction of the room did they plaster? d What fraction of the whole room did Jamie plaster? 17 1 mile = 13 5 km Lorna is driving from London to Edinburgh, a journey of 405 miles. a She passes a road sign which tells her she has 360 miles to go. How many kilometres is she from Edinburgh? b What fraction of the journey has she completed? 26.3 Multiplying mixed numbers ● I can multiply one mixed number by another To multiply two mixed numbers, you need to change them to improper fractions. Then you can multiply the numerators and multiply the denominators. Remember to simplify your answers. Develop fluency Work these out. a 2 3 of 3 1 2 a 2 3 of 3 1 2 = 2 3 × 7 2 Change 3 1 2 to an improper fraction. = 14 6 2 × 7 = 14 and 3 × 2 = 6 = 7 3 Simplify the fraction by dividing numerator and denominator by 2. = 21 3 Convert the answer to a mixed number. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 361 2867_P001_680_Book_MNPB.indb 361 24/03/2020 15:33 4/03/2020 15:33

362 Chapter 26 b 1 3 4 × 2 1 2 b 1 3 4 × 2 1 2 = 7 4 × 5 2 Change both mixed numbers to improper fractions. = 35 8 = 43 8 Convert the answer to a mixed number. 1 Work these out. a 1 2 × 1 1 2 b 1 2 × 1 1 4 c 1 4 × 2 1 2 d 1 4 × 3 1 2 2 Work these out. a 1 2 3 × 2 1 2 b 1 1 4 × 3 1 2 c 2 1 3 × 2 1 2 d 4 1 2 × 2 2 3 3 Multiply 2 2 3 by: a 1 2 3 b 2 1 3 c 1 3 4 d 5 1 2 4 Work these out. a ( 3 4) 2 b (21 4) 2 c (41 4) 2 d (13 4) 2 5 Work these out. a 2 2 3 × 3 3 4 b 3 3 4 × 3 1 5 c 1 2 5 × 2 4 7 d 2 4 9 × 4 1 3 6 Work these out. a 2 1 5 × 6 b 4 3 4 × 7 c 3 2 3 × 4 d 5 2 7 × 5 7 Work these out. a 2 3 × 3 5 × 1 2 b 1 2 3 × 3 5 × 4 c (2 1 2) 2 d ( 2 3) 3 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 362 2867_P001_680_Book_MNPB.indb 362 24/03/2020 15:33 4/03/2020 15:33

363 26.3 Reason mathematically Peter has been given this calculation to do: 3 3 5 × 4 2 3 Here is his working out: 3 3 5 × 4 2 3 = 3 × 4 3 5 × 2 3 = 12 6 15 = 72 15 = 4 4 5 However, the actual answer is 16 4 5. a Explain how you know, without making the calculation, that Peter has made a mistake. b What mistake has Peter made? a Looking at the whole-number parts, 3 and 4, the answer should be at least 3 × 4 =12, so an answer of 4 is obviously wrong. b Peter has used the wrong method. His method would work if he wanted to add the two fractions. In multiplication calculations, the mixed numbers cannot be separated into whole numbers and fractions. 8 Rana and Ana have each worked out the answer for 4 2 7 × 5 1 4 . Rana has an answer of 12 3 7 and Ana 22 13 28 . a Without doing the calculation whose answer is more likely to be correct? Why? b What is the correct answer? 9 a Work out the value of x2 when x is equal to: i 1 1 2 ii 1 1 3 iii 1 1 4 iv 1 2 5 b Which answer in part a is closest to 2? 10 a Work out the value of x2 when x is equal to: i 1 1 5 ii 1 1 6 iii 1 1 7 iv 1 1 8 b What number are the answers to this series getting closer to? c Will any value of x2 be less than 1? Why? 11 Does ( 3 4) 2 × (1 1 2) 2 × 22 have the same value as ( 3 4 × 1 1 2 × 2) 2 ? Explain your reasoning. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 363 2867_P001_680_Book_MNPB.indb 363 24/03/2020 15:33 4/03/2020 15:33

364 Chapter 26 Solve problems A half marathon is run over a distance of 13 1 10 miles. On one particular course, four drinks stations are located at 1 5, 2 5, 3 5 and 4 5 of the race distance. At what distances from the race start are the drinks stations? Station 1: 13 1 10 × 1 5 = 2 31 50 miles Station 2: 13 1 10 × 2 5 = 5 6 25 miles Station 3: 13 1 10 × 3 5 = 7 43 50 miles Station 4: 13 1 10 × 4 5 = 10 12 25 miles 12 In cricket each session usually lasts 2 hours. a A player who bats for 2 3 4 sessions has batted for how many minutes? b A bowler is used for a quarter of one session, 2 3 of the next and half of the final one. How many minutes was he bowling for? c Due to bad weather one session is extended to last 1 2 5 of its normal time. How long is that? 13 a Work these out. i 1 5 × 1 1 2 ii 1 5 × 2 1 2 iii 1 5 × 3 1 2 iv 1 5 × 4 1 2 b The multiplications in part a follow a pattern. Write down and calculate the next two terms in the pattern. 14 Konika is doing some design work. She shades all the squares in her first rectangle purple and then part of a second identical rectangle purple. a Counting one rectangle as a whole one, what fraction of the two rectangles has been shaded? b Konika decides to change a quarter of the shaded squares to yellow. What fraction of a rectangle is yellow? c What fraction is still purple? 15 a Work these out. i 1 1 2 × 2 3 ii 1 1 3 × 3 4 iii 2 1 2 × 2 5 iv 1 3 4 × 4 7 b Write down two more multiplications like those in part a. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 364 2867_P001_680_Book_MNPB.indb 364 24/03/2020 15:33 4/03/2020 15:33

365 26.4 26.4 Dividing fractions ● I can divide one fraction or mixed number by another To divide by a fraction, you invert it or ‘turn it upside down’. This means that you swap the numerator and the denominator. You then multiply by the new fraction. If the calculation includes mixed numbers, change them to improper fractions first. Develop fluency Work out: a 3 ÷ 2 3 b 2 3 ÷ 3 4 a 3 ÷ 2 3 = 3 1 × 3 2 Write 3 as 3 1 and invert 2 3 to get 3 2. = 3 3 1 2 = 9 2 = 4 1 2 b 2 3 ÷3 4 = 2 3 × 4 3 Invert 3 4 but do not change 2 3 = 8 9 Work these out. a 2 1 4 ÷ 1 1 3 b 1 1 2 ÷ 3 1 2 a 2 1 4 ÷ 1 1 3 = 9 4 ÷ 4 3 Write the mixed numbers as improper fractions. = 9 4 × 3 4 Invert 4 3 and multiply. = 27 16 = 1 11 16 Write the improper fraction as a mixed number. b 1 1 2 ÷ 3 1 2 = 3 2 ÷ 7 2 Write the mixed number as improper fractions. = 3 2 × 2 7 Invert 7 2 and multiply. = 6 14 = 3 7 Simplify the fraction as much as possible © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 365 2867_P001_680_Book_MNPB.indb 365 24/03/2020 15:33 4/03/2020 15:33

366 Chapter 26 Work these out. 1 a 3 ÷ 1 2 b 3 ÷ 1 3 c 3 ÷ 2 5 d 3 ÷ 3 5 2 a 2 ÷ 1 4 b 5 ÷ 2 3 c 12 ÷ 3 4 d 2 ÷ 5 12 3 a 1 2 ÷ 1 4 b 1 2 ÷ 1 3 c 1 2 ÷ 1 5 d 1 2 ÷ 1 8 4 a 1 2 ÷ 2 5 b 1 4 ÷ 3 4 c 2 5 ÷ 1 10 d 1 6 ÷ 2 3 5 a i 1 4 ÷ 1 3 ii 1 3 ÷ 1 4 b i 1 3 ÷ 1 6 ii 1 6 ÷ 1 3 6 a 2 1 2 ÷ 1 2 b 2 1 2 ÷ 1 4 c 2 1 2 ÷ 3 4 d 2 1 2 ÷ 5 8 7 a 1 1 2 ÷ 2 3 b 3 1 4 ÷ 3 4 c 4 1 2 ÷ 2 3 d 7 1 2 ÷ 3 5 8 a 4 1 2 ÷ 1 2 b 6 1 2 ÷ 1 1 2 c 1 1 2 ÷ 4 1 2 d 3 1 2 ÷ 4 1 2 9 a 2 1 2 ÷ 1 3 4 b 12 1 2 ÷ 7 1 2 c 3 1 2 ÷ 4 2 3 d 10 1 2 ÷ 4 10 a ( 2 5 ÷ 1 2) ÷ 1 3 b 2 5 ÷ ( 1 2 ÷ 1 3) Reason mathematically a Work out: i 2 5 ÷ 10 ii 10 ÷ 2 5 b Explain why these answers are different. a i 1 25 ii 25 b The first asks you to find 1 10 of 2 5 (because ‘÷ 10’ means 1 10), which is 1 10 × 2 5 = 1 25 . The second is asking, ‘How many 2 5 are there in 10?’, which is 25 because 25 × 2 5 = 10. 11 a Work out i 5 ÷ 1 2 and 1 2 ÷ 5 ii 4 ÷ 1 3 and 1 3 ÷ 4 iii 8 ÷ 1 10 and 1 10 ÷ 8 b What did you notice each time? c When you divide an integer by a fraction, is the answer bigger or smaller? d When you divide a fraction by an integer, is the answer bigger or smaller? 12 What do you notice about the pairs of answers in question 5? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 366 2867_P001_680_Book_MNPB.indb 366 24/03/2020 15:33 4/03/2020 15:33

367 26.4 13 a Calculate 5 1 4 ÷ 2 1 3. b Check your calculation in part a by multiplying the answer by 2 1 3. Solve problems Max walks 2 1 2 miles from home to school in 45 minutes. Kate cycles 3 1 2 miles from home to school in 28 minutes. a How many minutes does it take Max to walk 1 mile b How many minutes does it take Kate to cycle 1 mile? c How many times further away from school is Kate’s home than Max’s home? a Max: minutes for 1 mile = 45 ÷ 2 1 2 = 45 ÷ 5 2 = 45 × 2 5 = 90 5 = 18 b Kate: minutes for 1 mile = 28 ÷ 3 1 2 = 28 ÷ 7 2 = 28 × 2 7 = 56 7 = 8 c Kate’s house is 3 1 2 ÷ 2 1 2 times further from school than Max’s: 7 2 ÷ 5 2 = 7 2 × 2 5 = 14 10 = 1 2 5 14 Ali has just discovered that, at the equator, a point on the Earth’s surface is spinning at 1060 mph on average, which is 1 2 3 times faster than it would be in London. This is because London is further north and has a smaller circle to travel in the same time. a How fast are you spinning if you live in London? b Kieran says this should be in km/h, so it needs to be divided by 5 8. However, Hussain says that’s wrong and you should multiply by 1 3 5. Who is right and how fast is London spinning in km/h? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 367 2867_P001_680_Book_MNPB.indb 367 24/03/2020 15:33 4/03/2020 15:33

368 Chapter 26 15 A bucket contains 15 1 2 litres of milk. 6 small bottles each holding 7 8 litre are filled with milk from the bucket. The remaining milk is poured into bottles each holding 1 1 2 litres. How many of the 1 1 2 litre bottles can be filled? 16 A garden patio is being laid. Its length is 5 3 4 m and its width is 2 1 2 m. The tiles to be used are square with a side length 1 4 m . How many tiles will be needed to cover the patio? Now I can… add any two mixed numbers multiply two fractions divide one fraction or mixed number by another subtract any two mixed numbers multiply one mixed number by another © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 368 2867_P001_680_Book_MNPB.indb 368 24/03/2020 15:33 4/03/2020 15:33

369 27 Circles 27.1 Parts of a circle ● I can define a circle and name its parts A circle is a set of points that are all the same distance from a fixed point, called the centre. The centre of the circle is usually called O. Develop fluency a Draw a semicircle with diameter 5 cm. b Draw a quadrant of a circle with radius 30 mm. a Draw a line 5 cm long, for the diameter. Mark a dot at the midpoint, for the centre. Use compasses to draw the semicircle with radius 2.5 cm. b Draw a line 3 cm long, for the radius. Construct a right angle at one end and draw a 3 cm line, perpendicular to the original line. Use your compasses to draw the arc with radius 3 cm to create the quadrant. a b 1 a A circle has a radius of 6 cm. What is the length of its diameter? b A circle has a diameter of 30 m. What is the length of its radius? 2 i Measure the radius of each circle, giving your answer in centimetres. ii Write down the diameter of each circle. c O b O a O 3 Draw circles with these measurements. a radius = 2.5 cm b radius = 3.6 cm c diameter = 8 cm d diameter = 6.8 cm 4 a Draw a circle with radius 33 mm. b Draw a circle with diameter 9.2 cm. c Draw a semicircle with diameter 8 cm. d Draw a quadrant of a circle with radius 52 mm. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 369 2867_P001_680_Book_MNPB.indb 369 24/03/2020 15:33 4/03/2020 15:33

370 Chapter 27 5 Construct these diagrams. a b 6 a Look at the diagram and name the type of line. i AB ii AC iii DE iv CF b Name the type of shape. i ABCA ii ACFA iii ABCFA iv GABCG 7 Draw each of these shapes accurately. Use a ruler, compasses and a protractor. a b c d 60° 4 cm 4 cm 6 cm 3 cm 4 cm Concentric circles Semicircle Quadrant of a circle Sector of a circle 8 Draw each of these shapes accurately. a b c 6 cm 2 cm 2 cm 4 cm 4 cm 4 cm 8 cm A B E C F G © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 370 2867_P001_680_Book_MNPB.indb 370 24/03/2020 15:33 4/03/2020 15:33

371 27.1 9 Construct these diagrams accurately. a b 5 cm 3 cm 3 cm 4 cm c 10 Match each word on the left with its correct definition on the right a Circle A The distance from the centre of a circle to its circumference. b Arc B The distance across a circle, through its centre. c Semicircle C A set of points all the same distance from a fixed point. d Sector D A straight line that joins two points on the circumference of a circle. e Circumference E A straight line that touches a circle at only one point on its circumference. f Diameter F A portion of a circle that is enclosed by a chord and an arc. g Tangent G The length round the outside of the circle. h Radius H A portion of a circle enclosed by two radii and an arc. i Segment I A part of the circumference of the circle. j Chord J One half of a circle; either of the parts cut off by a diameter. Reason mathematically Draw a circle with centre O and a radius of 5 cm. Use a protractor to draw three radii that form angles of 120° at the centre of the circle, as shown in the diagram. Join the three points where the radii meet the circumference, to make a triangle. a Explain why the triangle is equilateral. b Explain how you could use a similar method to draw a square. a The three radii are all the same length, with the same angle, 120° between each radii. This gives us three congruent shapes. Each chord is the same length and so each side of the triangle will be the same, hence the triangle will be equilateral. b Draw a circle. Then draw four radii that form angles of 90° at the centre of a circle. Now join the four points where the radii meet the circumference to make a square. O © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 371 2867_P001_680_Book_MNPB.indb 371 24/03/2020 15:33 4/03/2020 15:33

372 Chapter 27 11 a Draw a circle with centre O and a radius of 4 cm. Use a protractor to draw six radii that form angles of 60° at the centre of the circle, as shown in the diagram. Join the six points where the radii meet the circumference, to make a regular hexagon. b Use a similar method to draw a regular pentagon. 12 Amber draws a circle. She uses a protractor to draw eight radii that form angles of 45° at the centre of the circle. She joins the eight points where the radii meet the circumference. a What shape has she drawn with these eight points? b Explain how you could use a similar method to draw a decagon. 13 Explain how you could use a circle to help you draw a regular twenty-sided polygon. 14 Theo draws a circle, then uses a protractor to draw radii that form angles of 20° at the centre of the circle. He says: ‘When I join the points where the radii meet the circumference, I will have drawn a regular polygon with 20 sides.’ Is Theo correct? Explain your answer. Solve problems Draw a circle. Then draw two tangents to the circle that meet each other at a point T. Label the points of the tangents that meet the circle A and B. Measure the lengths AT and BT. What can you say about these lengths? Measuring the two lengths AT and BT shows that they are equal. O A T B 15 a Draw a circle around a circular object so that you don’t know where the centre is. b Draw a chord AB on the circle. c Mark a point halfway along AB, label it X. d Draw a line perpendicular to AB at X so that the line passes though the circle. e What name can we give to the part of the perpendicular line that passes through the circle? 16 a Find a circular object with an unknown centre. Draw a circle around it. b Draw two chords AB and BD on the circle. c Mark a point halfway along each chord, label them X and Y. d Draw perpendicular lines to the chords at X and Y so that the lines pass right across the circle. e What is special about the point where these two perpendicular lines cut each other? O © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 372 2867_P001_680_Book_MNPB.indb 372 24/03/2020 15:33 4/03/2020 15:33

373 27.2 17 Draw a circle. Then draw two tangents to the circle that meet at a point T. Label the points of the tangents that meet the circle as A and B. a Draw perpendicular lines to the tangents at A and B. b Where do these two perpendicular lines meet? 18 Eve draws two tangents to a circle, centre O, that meet point T. The points of the tangents that meet the circle are A and B. What can you say about the triangles AOT and BOT? 27.2 Formula for the circumference of a circle ● I can calculate the circumference of a circle The formula for calculating the circumference, C, of a circle with diameter d is written as C = πd. As the diameter is twice the radius, r, the circumference is also given by the formula: C = πd = π × 2r = 2πr These formulae include a special number represented by the Greek letter π (pronounced pi). It is impossible to write down the value of π exactly, as a fraction or as a decimal, so you will use approximate values. The most common of these are: • π = 3.14 (as a decimal rounded to two decimal places) • π = 3.142 (as a decimal rounded to three decimal places) • π = 3.141 592 654 (on a scientific calculator) • π = 22 7 (as a fraction). Develop fluency Calculate the circumference of each circle. Give each answer correct to one decimal place. a 6 cm b 3.4 m O O a The diameter d = 6 cm, which gives: C = πd = π × 6 = 18.8 cm (to 1 dp) b The radius r = 3.4 m, so d = 6.8 m. This gives C = πd = π × 6.8 = 21.4 m (to 1 dp) In this exercise, take π = 3.14 or use the π key on your calculator. 1 Find the circumference of a circle with: a diameter 10 m b radius 4 m. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 373 2867_P001_680_Book_MNPB.indb 373 24/03/2020 15:33 4/03/2020 15:33

374 Chapter 27 2 Calculate the circumference of each circle. Give each answer correct to one decimal place. a b c d e 7 cm 11 mm 21 mm 2.4 m 1.4 cm O O O O O 3 Calculate the circumference of each wheel. Write your answers correct to the nearest centimetre. a b 60 cm 17.2 cm 4 Calculate the circumference of each coin. Give each answer to one decimal place. a b 21 mm Car wash coin 14 mm 5 Copy and complete the table for each circle (using suitable values of π). Radius Diameter Circumference 7 cm 5 cm 314 6 Calculate the perimeter, P, of this semicircle. Give your answer correct to one decimal place. 8 cm 7 Calculate the perimeter of this quadrant, taken from a circle radius 4 cm. O Give your answer to one decimal place. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 374 2867_P001_680_Book_MNPB.indb 374 24/03/2020 15:33 4/03/2020 15:33

375 27.2 8 A circle, centre O, radius 5 cm has three radii drawn as shown. Calculate the perimeter of the shape AOB, giving your answer to one decimal place. 9 Calculate the perimeter of this sector of a circle radius 14 cm. Give your answer as an integer. 10 The diagram is made up of a rectangle 20 cm by 100 cm with a semicircle of diameter 20 cm at each end. Calculate the perimeter of the shape, giving your answer to the nearest integer. Reason mathematically Mae told Ben the perimeter of her circular table is 66 cm. Ben said: ‘The diameter of the table will be 21 cm.’ Could Ben be correct? Explain what assumption Ben has made. Yes Ben is correct if he assumed π to be 22 7 , as 22 7 × 21 = 66 11 The Earth’s orbit can be taken as a circle with radius approximately 150 million kilometres. Calculate the distance the Earth travels in one orbit of the Sun. Give your answer correct to the nearest million kilometres. 12 The distance round a circular running track is 200 m. Calculate the radius of the track. Give your answer correct to the nearest metre. 13 The London Eye has a diameter of 120 m. How far would you travel in one complete revolution of the wheel? Give your answer correct to the nearest metre. 14 Calculate the total length of the lines in this crop circle diagram. It has two semicircles, a circle and straight lines. Write your answers correct to the nearest metre. O 120á 120á B A C O 20 cm 100 cm 15 m 10 m 10 m 15 m 35 m 50 m 5 m 5 m 12 m © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 375 2867_P001_680_Book_MNPB.indb 375 24/03/2020 15:33 4/03/2020 15:33

376 Chapter 27 Solve problems The diagram shows the dimensions of a running track at a sports centre. The bends at the ends are semicircles. Calculate the distance round the track. Give your answer to the nearest metre. The circumference of circle made up of the semi-circular ends is π × 51 = 160 m. Add the two straight sections to give 160 + 240 = 400 m 120 m 51 m 15 Calculate the total perimeter of this arched window. Give your answer correct to the nearest centimetre. 16 The radius of a quadrant is 20 mm. Calculate the perimeter in millimetres. Give your answer correct to three significant figures. 17 The curved parts of this shape are all semicircles. Calculate the perimeter of the shape. Give your answer correct to one decimal place. 18 The world’s tallest Ferris wheel is the Singapore Flyer, opened in 2008. Its diameter is 150 m and it has 28 capsules, spread equally around the circumference. Kamile gets into a capsule. Ewa gets into the next capsule. Find the distance travelled by Kamile before Ewa gets into her capsule, correct to the nearest metre. 40 cm 1.25 m 5 cm 5 cm 5 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 376 2867_P001_680_Book_MNPB.indb 376 24/03/2020 15:33 4/03/2020 15:33

377 27.3 27.3 Formula for area of a circle ● I can calculate the area of a circle The formula for the area, A, of a circle of radius r is A = πr2 . Develop fluency Calculate the area of each circle. Give your answers correct to one decimal place. a 3 cm b 3.4 m O O a The radius, r = 3cm, which gives: A = πr 2 = π × 32 = 9π = 28.3cm2 (to 1 dp) When using a calculator, you can use the ‘square’ key . Simply key in: b The diameter d = 3.4 m, so r = 1.7 m. This gives: A = πr2 = π × 1.72 = 2.89π = 9.1 m2 (to 1 dp) Note that you can also leave your answers in terms of π. For example, this may be necessary when you are not allowed to use a calculator. This gives answers of a 9π and b 2.89π. In this exercise, take π = 3.14 or use the π key on your calculator. 1 Find the area of a circle with: a diameter 10m b radius 4m. 2 Calculate the area of each of the following circles. Give each answer to one decimal place. a 8 cm b 7 cm c 9 cm d 3.6 cm O O O O © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 377 2867_P001_680_Book_MNPB.indb 377 24/03/2020 15:33 4/03/2020 15:33

378 Chapter 27 3 Calculate the area of each circle. Give your answers correct to one decimal place. a b c d e 1 cm 14 mm 2.1 m 3.5 cm 5.5 m O O O O O 4 Calculate the area of a circular tablemat with a diameter of 21cm. Give your answer correct to the nearest square centimetre. 5 A CD has a diameter of 12cm. Calculate its circumference and area. Give your answers to one decimal place. 6 The diagram is a quadrant of a circle with radius 5cm. Calculate the area of the quadrant. Give your answers to one decimal place. 7 This shape is a sector from a circle radius 9cm. Calculate the area of the shape. Give your answers to one decimal place. 8 Show that the area of this semicircle is 8πcm2 . 9 Calculate the area of a semicircle with radius 10 cm. Give your answer in terms of π. 10 Calculate the area of a quadrant with radius 12 cm. Give your answer in terms of π. Reason mathematically Laz has a circular plate with diameter 12 cm. His brother Theo has a square plate of side length 10 cm. Theo says to Laz: ‘My square plate has a larger area than your circular plate.’ Is Theo correct? Explain your answer. Laz’s circular plate has area of π × 62 = 3.14 × 36 = 113.04 cm2 Theo’s square plate has an area of 102 = 100 cm2 So, Theo is incorrect. 11 Ollie has a circular table with a diameter of 30cm. He says the area of this table is 707cm. Is Ollie correct? Explain your answer. O O 8 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 378 2867_P001_680_Book_MNPB.indb 378 24/03/2020 15:33 4/03/2020 15:33

379 27.3 12 Andrew has a pizza with diameter 14cm. He says the area of the top is 154 cm2 exactly. What assumption must he make to be correct? Explain your answer. 13 Randal is working out the area of this circle. This is his working. Area = π × d = π × 8 = 8π cm Explain why Randal’s working is wrong. Write down the correct answer to the problem. 14 Finlay has cut out a circle of radius 5 cm and Jackson has cut out a circle of radius 10 cm. Jackson says to Finlay, ‘My circle is twice the size of yours, so the area of my circle should be twice the area of yours.’ Calculate the areas of both circles, giving your answers in terms of π. Is Jackson correct? Give a reason for your answer. Solve problems Calculate the area of the shaded part of this shape. The whole circle has area π × 52 = 3.14 × 25 = 78.5 cm2 The circle in the middle has area π × 12 = 3.14 × 1 = 3.14 cm2 So, the area of the shaded part is 78.5 – 3.14 = 75.36 cm2 1 cm 5 cm 15 The diagram represents a sports ground. The curved ends are semicircles. Calculate the area of the sports ground. Give your answer correct to the nearest square metre. 16 The length of the minute hand on a clock is 24 cm. Calculate the area swept by the minute hand in: a 1 hour b 5 minutes c 1 minute. Give your answers in terms of π. 4 cm O 120 m 51 m © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 379 2867_P001_680_Book_MNPB.indb 379 24/03/2020 15:33 4/03/2020 15:33

380 Chapter 27 17 Calculate the area of this shape. Write your answer correct to the nearest square centimetre. 6 cm 10 cm 18 Find the area of this shape. Give your answer in terms of π. 10 cm 10 cm Now I can… define a circle calculate the circumference of a circle calculate the area of a circle name the parts of a circle © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 380 2867_P001_680_Book_MNPB.indb 380 24/03/2020 15:33 4/03/2020 15:33

381 28 Finding probabilities 28.1 Using probability scales ● I can use a probability scale to represent a chance When you do something such as rolling a dice, this is called an event. The possible results of the event are called its outcomes. For example, rolling a dice has six possible outcomes: scoring 1, 2, 3, 4, 5 or 6. You can use probability to decide how likely it is that different outcomes will happen. Equally likely outcomes are those that all have the same chance of happening. For example, when you roll a dice, there are six different possible outcomes. This is because it could land so that any one of its six numbers shows on top. The probability of an equally likely outcome is: P(outcome) = the number of ways that the outcome could occur the total number of possible outcomes Probabilities can be written as either fractions, decimals or percentages. They always take values from 0 to 1. The probability of an event happening can be shown on the probability scale. If one outcome is the absolute opposite of another outcome, such as ‘raining’ and ‘not raining’, then the probabilities of the two outcomes add up to 1. Similarly, if the outcomes of an event do not overlap the probabilities add up to 1, for example, if the result of a game is win, lose or draw, then P(win) + P(lose) + P (draw) = 1. Develop fluency a What is the probability of scoring a number less than 5 when you roll a dice? b What is the probability of not scoring a number under 5 when you roll a dice? a There are four possible outcomes that give you a number less than 5: 1, 2, 3 and 4. There are six different possible outcomes altogether, when you roll a dice: 1, 2, 3, 4, 5 and 6. So. P(rolling a dice and getting a number less than 5) is 4 6 = 2 3. b These two outcomes are the opposite of each other, so the probabilities add up to 1. P(number not less than 5) is 1 – 2 3 = 1 3. 1 A set of cards is numbered from 1 to 50. One card is picked at random. Give the probability that the number on it: a is even b has a 7 in it c has at least one 3 in it d is a prime number e is a multiple of 6 f is a square number. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Impossible Even Certain © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 381 2867_P001_680_Book_MNPB.indb 381 24/03/2020 15:33 4/03/2020 15:33

382 Chapter 28 2 Joe has 1000 tracks on his phone. He has: 250 tracks of pop 200 tracks of blues 400 tracks of country and western 100 tracks of heavy rock 50 tracks of quiet romantic. He sets the player to play tracks at random. What is the probability that the next track to play is: a pop b blues c country and western d heavy rock e quiet romantic f not heavy rock? 3 The probability of outcomes A, B, C and D are shown on the scale. Copy the scale and mark underneath it the probabilities of A, B, C and D not happening. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 A B C D 4 Copy and complete each of these tables. Outcome Probability of outcome occurring (P) Probability of event not occurring (1 − P) A 1 4 B 1 3 C 3 4 D 1 10 E 2 15 Outcome Probability of outcome occurring (P) Probability of event not occurring (1 − P) A 2 3 B 0.35 C 8% D 0.04 E 5 8 F 0.375 5 A spinner has four coloured sections: blue, red, green and yellow. The probability of landing on some of the sections is shown in the table. Colour Blue Red Green Yellow Probability 0.2 0.3 0.1 Work out the probability of landing on yellow. 6 In a bus station there are 24 red buses, 6 blue buses and 10 green buses. Work out the probability that the next bus to leave is: a green b red c red or blue d yellow e not green f not red g neither red nor blue h not yellow. 7 The probability of an egg having a double yolk is 0.009. What is the probability that an egg does not have a double yolk? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 382 2867_P001_680_Book_MNPB.indb 382 24/03/2020 15:33 4/03/2020 15:33

383 28.1 8 100 rings are placed in a box. Ten are gold, 20 are silver, 36 are plastic and the rest are copper. A ring is chosen at random. What is the probability that it is: a gold b not silver c copper d not plastic? 9 The diagram shows 12 dominoes. A domino is chosen at random. Work out the probability that it: a has a 4 b does not have a 3 c has a total over 5 d totals less than 10 e has a total of 9 f is a double totalling less than 5. Reason mathematically A bag contains 32 counters. Some are black and the others are white. The probability of picking a black counter is 1 4. Show that there are 24 white counters in the bag. The probability of picking a white counter is 1 – 1 4= 3 4. So, the number of white counters is 3 4 of 32 = 3 4 × 32 = 24, 10 In a raffle there are 2400 tickets. The probability that a ticket is a winning ticket is 1 100 . Show that there are 2376 losing tickets. 11 A child is asked to choose a number at random from: 1 2 3 4 5 6 7 8 9 For each part, compare the probabilities, stating which, if any, is more likely. a even number or number more than 6 b prime number or odd number c multiple of 5 or multiple of 4 d triangular number or square number. 12 In a bag there are green, blue, red and yellow discs. Jack says this means the probability of picking a green disc is 1 4 because green is one of out of four colours. Is his statement correct? Give a reason for your answer. 13 A box contains blue and black pens. A pen is picked at random. There are more blue pens than black pens. What does this tell you about the probability a choosing a blue pen? Alternatively: Number of black counters is 1 4 of 32 = 1 4 × 32 = 8 So, the number of white counters is 32 – 8 = 24 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 383 2867_P001_680_Book_MNPB.indb 383 24/03/2020 15:33 4/03/2020 15:33

384 Chapter 28 Solve problems A bag contains 10 blue and 15 red discs. a A disc is picked at random and then replaced. What is the probability the disc is blue? b More red discs are added to the bag. The probability of picking a blue disc is now 1 3. How many red discs were added to the bag? a P(blue) is 10 25 = 2 5. b More red discs are added so the total number in the bag will increase so that 10 total = 1 3. So, the total number must be 30, meaning 5 red discs were added. 14 There are two white counters and one black counter in a bag. a One counter is picked and replaced. What is the probability that it is a black counter? b More black counters are put in the bag. The probability that a black counter is picked at random is now 2 3. How many more black counters were put in the bag? 15 In a box of cereal there is a free gift of a model dinosaur. There are five animals to make up the set: Tyrannosaurus Rex, Stegosaurus, Triceratops, Brachiosaurus, Diplodocus. You cannot tell which animal will be in the box. Each one is equally likely. a What is the probability that the animal is a diplodocus? b What is the probability that the animal is a stegosaurus or a triceratops? The box is opened. The animal is not a brachiosaurus. c Now what is the probability that the animal is a diplodocus? d Now what is the probability that the 1 is a stegosaurus or a triceratops? 16 The probability of Bel winning a match is 1 2. She says that this means the probability of losing is also 1 2. What assumption has she made? 28.2 Mutually exclusive outcomes ● I can recognise mutually exclusive outcomes Mutually exclusive outcomes cannot occur together. Each excludes the possibility of the other happening. For example, when you roll a dice, throwing a 1 or a 6 are mutually exclusive as you cannot get both results on the same throw of one dice. However, suppose you have a dice and are trying to throw numbers less than 4, but you also want to score an even number. Which number is common to both outcomes? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 384 2867_P001_680_Book_MNPB.indb 384 24/03/2020 15:33 4/03/2020 15:33

385 28.2 ● The numbers less than 4 are 1, 2 and 3. ● The even numbers are 2, 4 and 6. Because the number 2 is in both groups, the outcomes are not mutually exclusive. This means it is possible to achieve both outcomes at the same time if you throw a 2. You can use a Venn diagram to illustrate this. The two circles show the sets of the possible outcomes: ‘less than 4’ and ‘even’. In this diagram: ● the set ‘less than 4’ has 1, 2 and 3 in it ● the set ‘even numbers’ has 2, 4 and 6 in it. The numbers that satisfy either or both outcomes together are in the union of the sets. The numbers that can satisfy either outcome are in the intersection of the sets. Because there are some numbers in the intersection, this shows that the outcomes are not mutually exclusive. Notice that all the possible outcomes of the dice roll are included in the Venn diagram. The number 5 does not form part of either outcome, so it is positioned outside the circles. Develop fluency Liz is buying fruit. Here is a list of possible outcomes. A: She chooses strawberries. B: She chooses red fruit C: She chooses green apples D: She chooses red apples E: She chooses oranges She chooses one item only. State which pairs of outcomes are mutually exclusive. a A and B b A and E c B and C d B and D a Strawberries are red fruit, so they are not mutually exclusive. b Strawberries are not oranges, so they are mutually exclusive. c Green apples are not red fruit, so they are mutually exclusive. d Red apples are red fruit, so they are not mutually exclusive. 1 These are the home shirt colours of the teams in a football league: 7 are red, 4 are blue, 3 are white, 1 is blue and white stripes and 1 is black and white stripes. a Are red shirts and blue shirts mutually exclusive? b Are striped shirts and shirts containing blue mutually exclusive? 2 Which of these are mutually exclusive? A: Picking an odd number B: Picking a prime number C: Picking an even number 3 Which of these are mutually exclusive? A: Picking a square number B: Picking a prime number C: Picking a triangular number Less than 4 Even 1 3 2 4 6 5 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 385 2867_P001_680_Book_MNPB.indb 385 24/03/2020 15:33 4/03/2020 15:33

386 Chapter 28 4 A card is picked at random from a normal pack of cards. A pack has two red suits and two black suits. Each suit contains 13 cards, ranging from values 1 (ace) to 10 and then jack, queen and king, known as picture cards. a Is picking a red card and picking a picture card mutually exclusive? b Is picking a picture card and picking a number card mutually exclusive? c Is picking a number above 6 and picking a black card mutually exclusive? 5 In a game you need to roll a dice and score an odd number larger than 2. a Draw a Venn diagram showing the two sets ‘odd numbers’ and ‘numbers larger than 2’. b Are the outcomes ‘scoring an odd number’ and ‘scoring a number larger than 2’ mutually exclusive? c Use your Venn diagram to state the probability of rolling an odd number larger than 2. 6 Soolin has a bag containing ten cards, each showing one of the integers 1 to 10. She is playing a game and needs to select a card at random that shows a prime number smaller than 4. a Draw a Venn diagram showing the two sets ‘prime numbers’ and ‘numbers smaller than 4’. b Are the outcomes ‘selecting a prime number’ and ‘selecting a number smaller than 4’ mutually exclusive? c Use your Venn diagram to state the probability of selecting a card showing a prime number smaller than 4. 7 A number square contains the numbers from 1 to 100. Numbers are chosen from the number square. Here is a list of outcomes. A: The number chosen is greater than 50. B: The number chosen is less than 10. C: The number chosen is a square number (1, 4, 9, 16, …). D: The number chosen is a factor of 100 (1, 2, 5, 10, …). E: The number chosen is a prime number (2, 3, 5, 7, …). State whether the outcomes in each pair are mutually exclusive or not. a A and B b A and C c B and C d B and D e C and D f C and E 8 The diagram shows sketches of eight faces. Which pairs of outcomes are mutually exclusive? a Having a smiling face and a sad face. b Smiling and having both eyes shut. c Wearing a hat and having both eyes open. d Having both eyes open and a sad face. e Wearing a hat and having one eye shut. f Smiling and having one eye shut. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 386 2867_P001_680_Book_MNPB.indb 386 24/03/2020 15:33 4/03/2020 15:33

387 28.2 Reason mathematically a Draw a Venn diagram with the two sets ‘square numbers up to 50’ and ‘numbers less than 10’. b Which numbers are in both sets? c Are the outcomes ‘selecting a square number’ and ‘selecting a number smaller than 10’ mutually exclusive? Explain your answer. a Square numbers up to 50 Numbers less than 10 16 25 36 49 4 1 9 5 6 8 7 3 2 b 1, 4 and 9 are in both sets. c Not mutually exclusive as 1, 4 and 9 are in both sets. 9 In a game you need to roll a dice and get an odd number less than 5 to win. a Draw a Venn diagram for the two sets ‘odd numbers’ and ‘numbers less than 5’. b Are the outcomes ‘rolling an odd number’ and ‘rolling a number less than 5’ mutually exclusive? Explain your answer. 10 Give an example to show that multiples of 5 and multiples of 7 are not mutually exclusive. 11 a Copy and complete the table to show all the possible pairs of scores if you spin these two spinners. Spinner 1 Spinner 2 Total score +20 2 +2 −1 1 b Which two outcomes are mutually exclusive? Give a reason for your answer A: A score of +4 on the 1st spinner B: A score of −1 on the 2nd spinner C: A total score of 1 D: A total score of 3 12 These are some outcomes from rolling a dice. A: Rolling a 5 or a 6 B: Rolling an odd number C: Rolling a 4 or a 6 D: Rolling a number less than 6 E: Rolling a number greater than 1 a Which two outcomes are mutually exclusive? Give a reason for your answer. b For each of the other pairs of outcomes, give a reason why they are not mutually exclusive. +4 +2 –3 +1 0 –1 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 387 2867_P001_680_Book_MNPB.indb 387 24/03/2020 15:33 4/03/2020 15:33

388 Chapter 28 Solve problems An ordinary six-sided dice is rolled. One possible outcome is scoring an odd number. List three different outcomes that are mutually exclusive to getting an odd number. For example: an even number, a 2, a number greater than 5. 13 Ten cards are numbered 1 to 10 A card is picked at random. One possible outcome it to pick the card with 5 on it. List three different outcomes that are mutually exclusive with picking the card with 5 on it. 14 A spinner has three sections red, blue and yellow. There are four possible outcomes. A: Landing on red B: landing on blue C: landing on yellow D: Not landing on red How many pairs of these outcomes are mutually exclusive? 15 A dice has six faces numbered 1, 2, 3, 4, 5 and 6. Here are four possible outcomes. A: Rolling a 1 B: Rolling a 6 C: Rolling a square number D: Rolling a prime number How many pairs of these outcomes are mutually exclusive? 28.3 Using sample spaces to calculate probabilities ● I can use sample spaces to calculate probabilities To help you work out the probabilities of outcomes happening together you can use a table or diagram called a sample space, which is the set of all possible outcomes from a specific event. You can use a Venn diagram or a table to illustrate a sample space. Develop fluency Find the probability of getting a head and a 6 when you roll a dice and flip a coin at the same time. This sample space shows all the possible outcomes of flipping a coin and rolling a dice at the same time. You can now work out the probability of getting both a head and a 6. P(outcome) = the number of ways that the outcome could occur the total number of possible outcomes P(head and a six) = 1 12 123456 Head H, 1 H, 2 H, 3 H, 4 H, 5 H, 6 Tail T, 1 T, 2 T, 3 T, 4 T, 5 T, 6 1 2 3 4 5 6 7 8 9 10 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 388 2867_P001_680_Book_MNPB.indb 388 24/03/2020 15:33 4/03/2020 15:33

389 28.3 In a class of 32 students, there are 15 boys. Out of the 8 left-handed students in the class, 6 are girls. What is the probability of choosing a right-handed boy from the class? You could put this information into a table. Boy Girl Total Left-handed 2 6 8 Right-handed 13 11 24 Total 15 17 32 You can now see that: P(right-handed boy) = 13 32 You can also show this in a Venn diagram. 13 Boys (15) Left-handed (8) Girls (17) 2 6 11 Pupils (32) 1 a Draw a sample space to show the results of flipping a coin and rolling a dice at the same time. b Use your sample space to find the probability of getting: i a 3 and a tail ii a head and an even number iii a number less than 5 and a tail. 2 Two tetrahedral (four-sided) dice, each numbered 1, 2, 3 and 4, are rolled together. Use a sample space diagram to show all the possible totals when the numbers are added. 3 Two children, Kim and Franz, have a bag containing 1p, 2p, 5p, 10p, 20p, 50p and £1 coins. They each write their name on a card and put the cards in the same bag. A card and a coin are taken from the bag at random. The coin is given to the named person. a Make a table to show the possible outcomes. b Calculate the probability that: i Kim receives 20p ii Franz receives less than 10p iii one of them receives 10p iv Kim does not receive £1 v neither receives more than 20p. 4 A market trader sells jacket potatoes plain, with cheese or with beans. Clyde and Delroy each buy a jacket potato. a Copy and complete the sample space table. b Write down the probability of: i Clyde choosing plain ii Delroy choosing plain iii both choosing plain iv Clyde choosing plain and Delroy choosing beans v Clyde choosing beans and Delroy choosing cheese vi both choosing the same vii neither choosing plain viii each choosing a different flavour. Clyde Delroy plain plain plain cheese © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 389 2867_P001_680_Book_MNPB.indb 389 24/03/2020 15:33 4/03/2020 15:33

390 Chapter 28 5 Bret rolls two dice and adds the scores together. Copy and complete the sample space of his scores. a What is the most likely total? b Write down the probability that the total is: i 4 ii 5 iii 1 iv 12 v less than 7 vi less than or equal to 7 vii greater than or equal to 10 viii even ix 6 or 8 x greater than 5. 6 Bart rolls two dice and then multiplies the results together to give a score. a Draw the sample space of his scores. b Which is the most likely to occur? c What is the probability of rolling a score greater than 17? 7 An ordinary dice is rolled at the same time as this spinner is spun. Their numbers are added together to give the total score. a Make a list of all the possible outcomes. (Axes like the set to the right may help.) b How many possible outcomes are there? c Find the probability of scoring: i 8 ii 1 iii an even number iv any score other than 5 v a score higher than 6 vi a total where the dice score is lower than the spinner score. Reason mathematically Tia throws two coins. Gary says: ‘There are three possible outcomes: 2 heads, head and tail or 2 tails, so the probability of 2 heads is 1 3.’ Is he correct? Give a reason for your answer. Listing the outcomes in a table shows that there are four outcomes, so the probability of 2 heads is 1 4. He is incorrect. 2nd coin Head Tail 1st coin Head HH HT Tail TH TT 8 Three coins are thrown. By listing all possible outcomes show that the probability of three heads is 1 8. The first three have been done for you. HHH HHT HTH 123456 123 2 3 5 2 3 1 4 Score on dice 4 5 3 2 1 123456 Score on spinner © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 390 2867_P001_680_Book_MNPB.indb 390 24/03/2020 15:33 4/03/2020 15:33

391 28.3 9 A bag contains apples, bananas and pears. Liam chooses two fruits at random. a List all the possible outcomes. b You are told that you have more chance of choosing an apple and pear than an apple and a banana. Explain how that could happen. 10 A combination lock has four digits. Each digit is 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. Mark remembers that: • all the digits are different • all the digits are odd • the first digit is 3 and the last digit is 7. Show that there are six possible combinations. Solve problems At a school fair there is a competition. a Draw a sample space diagram to show all the possible outcomes. b What is the probability of winning a prize? a 123456 1D××××× 2×D×××× 3××D××× 4×××D×× 5××××D× 6×××××D b P(double) is 6 36 = 1 6 Roll a double with 2 dice to win a prize. 11 Bella spins two spinners together. Each spinner has the numbers 0, 1, 2 and 3 on it. The numbers the spinners land on are added to give a score. a Draw a sample space diagram to show all the possible scores. b What is the probability that the score is less than 2? c What assumption have you made in part b? 12 A café makes 100 paninis. All the paninis are either meat or cheese, but 75 of them have salad in them, as well. 30 paninis have cheese with salad. 15 paninis have meat with no salad. Work out the probability of selecting at random a cheese panini without salad. 13 Two piles of three cards are each numbered 1 to 3. One card from each pile is turned over and the numbers are multiplied to give a score. Which is more likely; an odd score or an even score? Now I can… use a probability scale to represent a chance recognise mutually exclusive outcomes use sample spaces to calculate probabilities © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 391 2867_P001_680_Book_MNPB.indb 391 24/03/2020 15:33 4/03/2020 15:33

392 29 Equations and formulae 29.1 Equations with and without brackets ● I can solve equations involving brackets If an equation involves brackets you can either multiply out the brackets or divide each side first, to remove the need for the brackets. Develop fluency Solve these equations. a 3x – 2 = 15 b 3(x – 2) = 15 c 1 3 (x – 2) = 15 a 3x – 2 = 15 3x = 17 Add 2 to both sides. 15 + 2 = 17 x = 17 3 Divide both sides by 3. = 5 2 3 17 ÷ 3 = 5 remainder 2. Write the answer as a mixed number, because the decimal for 5 2 3 is 5.666 666… and you would need to round it. b 3(x – 2) = 15 There are two ways to solve this. Method 1 3(x – 2) = 15 x – 2 = 5 Divide both sides by 3. 15 ÷ 3 = 5 x = 7 Add 2 to both sides. Method 2 3(x – 2) = 15 3x – 6 = 15 Multiply out the brackets. 3(x – 2) = 3x – 6 3x = 21 Add 6 to both sides. x = 7 Divide by 3: 21 ÷ 3 = 7 Make sure you can use both of these methods. c 1 3 (x – 2) = 15 Finding 1 3 is the same as dividing by 3. x – 2 = 45 Multiply both sides by 3. 15 × 3 = 45 x = 47 Add 2 to both sides. 45 + 2 = 47 The equation in part c could also be written as x − 2 3 = 15 and solved. Notice that multiplying out the brackets of 1 3 (x – 2) = 15 first would give 1 3 x – 2 3 = 15. Then you would have to deal with coefficients that are fractions. The method shown above is usually easier. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 392 2867_P001_680_Book_MNPB.indb 392 24/03/2020 15:33 4/03/2020 15:33

393 29.1 1 Copy and complete these equations, given that x = 7. a 4x + 5 = . . . b 4(x + 5) = . . . c 21(x + 5) = . . . d x + 5 3 = . . . 2 Solve these equations. If the solution is not a whole number, write it as a mixed number. a 2x – 12 = 14 b 5y + 4 = 49 c 5k – 11 = 2 d 8n + 3 = 40 3 Solve these equations. a 2(x – 3) = 16 b 4(x + 5) = 32 c 3(y – 6) = 36 d 3(a – 4) = 120 4 Solve these equations. a 1 4 x = 12 b 1 4 x + 3 = 12 c 1 4 (x + 3) = 12 d x + 4 12 = 12 5 Solve these equations. a 1 2 y – 5 = 7 b 1 2 t + 4 = 13 c 8 = 1 2 (t – 11) d 1 8 (x + 10) = 4 6 Look at this equation: 3(x – 5) = 11 a Solve the equation by multiplying out the brackets first. b Solve the equation by dividing by 3 first. 7 Solve these equations. Give the answers as mixed numbers. a 4x – 12 = 15 b 4(x – 3) = 9 c 2(y – 11 4) = 11 d 10r + 5 = 69 8 Solve these equations. Write the answers as decimals. a 2x + 4.1 = 11.3 b 1.4(w + 6.2) = 12.6 c t − 1.7 3.2 = 4.5 d 1.8t + 32 = 144.5 9 Solve these equations. a a − 2 4 = 3 b b + 6 2 = 5 c c + 9 11 = 4 d d − 4 7 = 5 10 Solve these equations. Write the answers as decimals. Do not use a calculator. a 3.6p + 2.8 = 17.56 b 2q − 1.4 3 = 2.8 c 5(w + 3.2) = 29.5 Reason mathematically This is Johnny’s homework. Explain what is wrong with each of his solutions. a 8x – 5 = 19 8x = 19 – 5 = 14 x = 14 ÷ 8 = 1.75 b 3(5x – 1) = 27 15x – 1 = 27 15x = 30 x = 2 a Second line error, should be 8x = 19 + 5 = 24 x = 24 ÷ 8 = 3 b Second line error, should be 5x – 1 = 27 ÷ 3 = 9 5x = 9 + 1 = 10 x = 2 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 393 2867_P001_680_Book_MNPB.indb 393 24/03/2020 15:33 4/03/2020 15:33

394 Chapter 29 11 This is Helmut’s homework. Find any errors and explain what is wrong. a 9x – 2 = 25 9x – 2 = 25 + 2 = 3 b 3 + 4y = 19 3 + 4y = 19 4y = 19 + 3 = 22 y = 5.5 c 2(2x – 3) = 14 2(2x – 3) = 14 4x – 3 = 14 ⇒ 4x = 17 x = 4.25 12 This is Jos’s homework. Find any errors and explain what is wrong. a 4x + 3 = 9 4x = 9 + 3 = 12 x = 12 ÷ 4 = 3 b 4(2x + 3) = 24 8x + 8 = 24 8x = 16 x = 2 13 Ann helped her sister with this question. Solve the equation 2(b + 2) 5 = 4. Here is her solution 2(b + 2) = 9 Divide both sides by 2 b + 2 = 4.5 So b = 2.5 Is Ann correct? Explain your answer. 14 Here is Peter’s homework. He was asked to solve the equation 6x − 3 3 = 3. 6x – 3 = 9 6x = 12 x = 2 Is Peter correct? Explain your answer. Solve problems The perimeter of this rectangle is 15 cm. a Write down an equation to show this. b Solve the equation to find the value of x. a x + x + 2(x + 1) + 3(x – 5) = 15 7x – 13 = 15 b 7x = 28 x = 4 2(x+1) 3(x–5) x x 15 The perimeter of this trapezium is 371 2 cm. a Write down an equation to show this. b Solve the equation to find the value of x. 16 The formula for the perimeter of a rectangle is given by P = 2(a + b). a and b are different positive integers. a Write down all the possible pairs of values for a and b when P = 16. b Write down all the possible values for P when the area of the rectangle is 30. c Find the value of a when P = 356 and b = 29. d Explain why the rectangle cannot have a perimeter of 35. x 12 cm 1 2 x x © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 394 2867_P001_680_Book_MNPB.indb 394 24/03/2020 15:33 4/03/2020 15:33

29.2 17 The perimeter of this pentagon is 24 cm. 4y“ 2y y y 4y“ 18 The perimeter of this triangle is 11 cm. 3x“ 2x + 1 x 29.2 Equations with the variable on both sides ● I can solve equations with the variable on both sides Some equations have the unknown variable on one side only. It will often happen that the variable appears on both sides of the equation and then you need to bring the variables all to the same side in order to combine them. Develop fluency Jake is thinking of a number, ‘If I add 12 to my number it is the same as tripling it and subtracting 45.’ What is Jake’s number? Call Jake’s number x. Then you can write this equation: x + 12 = 3x – 45 This equation has x on both sides. When the variable occurs on both sides of the equal sign, remove it from one side by adding or subtracting it on the other side. x + 12 = 3x – 45 First, subtract x from both sides. 12 = 2x – 45 This removes the x from the left-hand side. On the right, 3x – x = 2x. Now x only appears on one side. Solve this in the usual way. 57 = 2x Add 45 to both sides. 12 + 45 = 57 28.5 = x Divide by 2. Half of 57 is 28.5. Solve these equations. a 2x – 4 = 5x – 24 a 2x – 4 = 5x – 24 Subtract 2x from both sides. –4 = 3x – 24 You now have –4 on the left-hand side. Add 24 to both sides. 20 = 3x –4 + 24 = 20. Now divide by 3. x = 20 3 = 6 2 3 It is better to leave the answer as a fraction in this case. b 4y + 8 = 43 – y b 4y + 8 = 43 – y To remove the –y from the right-hand side, add y to both sides. 5y + 8 = 43 4y + y = 5y. Solve this new equation in the usual way. 5y = 35 Subtract 8 from both sides. 43 – 8 = 35 y = 7 Divide by 5 to find the value of y. You can check this is correct. When y = 7, then 4y + 8 = 4 × 7 + 8 = 36 and 43 – y = 43 – 7 = 36. They have the same value. a Write down an equation to show this. b Solve the equation to find the value of x. a Write down an expression to show this. b Solve the equation to find the value of y. 395 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 395 2867_P001_680_Book_MNPB.indb 395 24/03/2020 15:33 4/03/2020 15:33

396 Chapter 29 Solve these equations. 1 a 2x = x + 15 b 4x = x + 45 c 3t = t + 24 d 5x = x + 44 2 a 4x = 2x + 18 b 5y = 2y + 18 c 6t = 2t + 18 d 6k = 5k + 18 3 a 2y – 8 = y + 12 b 4z – 17 = z + 4 c 3x – 14 = x + 8 d 6x – 13 = 2x + 9 4 a m = 3m – 16 b n = 5n – 84 c 2p = 5p – 18 d 4x = 10x – 48 5 a 2x – 15 = x + 3 b 3t – 7 = t + 14 c 5x – 4 = 2x + 6 d n + 15 = 2n – 19 e 3 + 4x = 17 + 2x f 5d – 27 = 3d – 1 g x = 5x – 20 h 2.5x + 6 = 3x – 4 6 a x = 24 – x b 2x = 24 – x c 3x = 24 – x d 5x = 24 – x 7 a x – 12 = 18 – x b 2x – 5 = 22 – x c 4x + 3 = 27 – 2x d 40 – 2x = 4 + x e 3x – 1 = 19 – 2x f x + 28 = 100 – 2x g 6x – 4 = 10 – x h 31 – 21 2x = 16 + 1 2x 8 a x + 11 = 3x + 2 b x + 2 = 11 – 3x c x + 2 = 3x – 11 d 11 – x = 3x – 2 9 a 8y = 6y – 10 b 6k – 6 = 9k c –4r = 3r + 21 d 7n – 3 = 3n – 15 10 a 6 + 2x = x – 5 b 6 + x = 5x – 2 c 5x + 2 = 6 – x d 2 + x = 5 – 6x Reason mathematically This is David’s homework. Is he correct? Explain your answer. Solve the equation 5x + 2 = 3x + 8. 5x + 2 = 3x + 8 5x – 3x = 8 + 2 2x = 10 x = 5 The second line is incorrect, it should be 5x – 3x = 8 – 2 giving 2x = 6 x = 3 11 This is Joel’s homework. Is Joel correct? Explain your answer. 4x – 3 = x + 9 4x – x = 9 – 3 = 6 3x = 6 x = 2 12 This is Amy’s homework. Explain what is wrong with each part. a 4x = 9 – 2x 4x – 2x = 9 2x = 9 b 3(x – 3) = 4x 3x – 9 = 4x 9 = x © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 396 2867_P001_680_Book_MNPB.indb 396 24/03/2020 15:33 4/03/2020 15:33

397 29.2 13 Kathy helped her brother with this question. Solve the equation b + 8 3 = b. Here is her solution: b + 8 = 3b b + 3b = 8 4b = 8 b = 2 Is she correct? Explain your answer 14 Here is Jack’s homework. Solve 8x − 5 2 = 6x. 8x – 5 = 6x 8x – 6x = 5 x = 3 Is Jack correct? Explain your answer. Solve problems Find the area of this rectangle. The top length is equal to the bottom length, leading to the equation: 5x – 1 = 3x + 4 5x – 3x = 4 + 1 2x = 5 x = 2.5 Area of rectangle = height × base = (2 × 2.5 – 1) × (3 × 2.5 + 4) = 4 × 11.5 = 46 square units 5x–1 2x–1 3x+4 15 The perimeters of these shapes are all the same length. a Write an equation to show this. b Solve the equation. c Work out the perimeter of each shape. 16 Simrath gets £8 pocket money per week. Ruby gets £10 pocket money per week. Simrath spent £x of her pocket money on stamps. a Write an expression for the amount of money Simrath had after she bought her stamps. Ruby bought five times as many stamps as Simrath. b Write an expression for the amount of money Ruby had after she bought her stamps. Simrath and Ruby each had the same amount of money left after buying their stamps. c Write and solve an equation to find out how much a stamp cost. d How much money did Ruby have left after buying her stamps? 17 a Find the area of this rectangle. b Find the area of this square. x 12 x x 35 x x x + 2 7x – 9 4x + 3 2x + 1 22 – x © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 397 2867_P001_680_Book_MNPB.indb 397 24/03/2020 15:33 4/03/2020 15:33

398 Chapter 29 29.3 More complex equations ● I can solve equations with fractional coefficients ● I can solve equations with brackets and fractions When you have brackets in an equation where the variable occurs more than once, it is usually best to multiply out the brackets first. When there is a fraction in front of a bracketed term, you can get rid of it by multiplying by the denominator. Develop fluency Solve the equation 3(t – 11) = 2(t + 8). 3(t – 11) = 2(t + 8) There are two sets of brackets. Multiply out both of them. 3t – 33 = 2t + 16 Now subtract 2t from both sides. t – 33 = 16 By subtracting 2t you remove the t-term from one side. t = 49 Add 33 to both sides. 16 + 33 = 49 You can check this is correct. 3(49 – 11) = 3 × 38 = 114 and 2(49 + 8) = 2 × 57 = 114 so they are the same. Solve the equation 2 3(x + 8) = x. 2 3(x + 8) = x First, multiply by 3. 2(x + 8) = 3x Now multiply out the brackets. 2x + 16 = 3x Now subtract 2x from both sides. 16 = x 3x – 2x = x Solve these equations. 1 a 2(x – 4) = 10 b 2(x – 4) = x c 2(x – 4) = x + 10 d 2(x + 4) = x + 10 2 a 3(y – 5) = 2y + 9 b a + 15 = 3(a – 1) c 2(t + 12) = 5t d 18 + x = 3(x – 8) e 2(n + 6) = 3n – 10 f 3x = 2(20 – x) 3 a 5x + 2 = 20 b 5(x + 2) = 20 c 5(x + 2) = x + 20 d 5(x – 2) = 20 – x 4 a 3(x + 2) = 2x + 8 b 3(2g + 3) = 4g + 17 c 8w – 12 = 3(3w – 1) 5 a 3(x – 6) = 2(x + 3) b 2(a + 9) = 3(a – 1) c 4(t – 3) = 2(t + 8) d 4(10 – p) = 2(5 + p) e 10(x + 4) = 2(x + 20) f 5(x – 4) = 3(x + 7) 6 The answers are the whole numbers from 1 to 8. Each answer should be used once. a 2(a + 6) = 3(a + 3) b 4(b – 1) = 2(b + 6) c 5(c + 2) = 3(c + 6) d 2(d + 2) = 4(d – 2) e 6(8 – e) = 12(e + 1) f 11(f + 3) = 22(f – 2) © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 398 2867_P001_680_Book_MNPB.indb 398 24/03/2020 15:33 4/03/2020 15:33

399 29.3 7 a 6(x + 3) = 10(x + 1) b 2(x – 1) = 5(x – 7) c 2(13 – x) = 4(8 – x) d 3(x + 6) = 4(x + 3) e 8(x + 10) = 12(x + 8) f 5(x + 11) = 10(x + 7) 8 a 2 5(w – 4) = 6 b x − 5 4 = x – 11 c 1 2(y + 9) = 2(y – 6) 9 a x − 5 3 = 2 b y + 8 4 = 5 c t + 8 2 = 6 d x + 1 2 = 12 e n − 4 4 = 7 f x − 12 8 = 2 10 a 3x − 2 2 = x b 3y + 1 2 = y c 5t − 3 2 = t Reason mathematically a Copy and complete this table. x 12345 4(x + 6) 8(9 – x) b Use the table to solve the equation 4(x + 6) = 8(9 – x). c Solve the equation algebraically and check that you get the same answer. a b From the table, both expressions have the same value when x = 4. c Multiply both brackets out first. 4x + 24 = 72 –8x 4x + 8x + 24 = 72 Add 8x to both sides 12x = 48 Subtract 24 from both sides and combine the xs. So x = 4, which is the same solution we had from the table. x 12345 4(x + 6) 28 32 36 40 44 8(9 – x) 64 56 48 40 32 11 a Copy and complete this table. x 56789 2(x + 1) 12 3(x – 2) 9 b Use the table to solve the equation 2(x + 1) = 3(x – 2). c Solve the equation algebraically and check that you get the same answer. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 399 2867_P001_680_Book_MNPB.indb 399 24/03/2020 15:33 4/03/2020 15:33

400 Chapter 29 12 Each of the sides of a hexagon is (y + 6) metres long. Each of the sides of a pentagon is (y + 8) metres long. The hexagon and pentagon have the same perimeter. a Write expressions for the perimeter of each shape. b Solve an equation to find the value of y. c Work out the perimeter of the hexagon. 13 Pair up the equations that have the same answers. a 3(w + 6) = 9(w – 2) b 3(w + 3) = 6(w – 6) c 6(w + 2) = 2(w + 11) d 5(10 – w) = 10(w – 4) e 9(w – 1) = 3(w + 2) f 10(22 – w) = 2(w + 20) Solve problems Ciara is c years old. She is six years older than her sister. In two years’ time she will be three times as old as her sister. How old is Ciara? Ciara’s sister is (c – 6) years old. Write what you know about, in two years’ time, as an equation. c + 2 = 3(c – 6) c + 2 = 3c – 18 Subtract c from both sides: 2 = 2c – 18 Add 18 to both sides: 20 = 2c So c = 10. Ciara is ten years old. 14 Here are two straight lines. The first line is divided into five equal parts, each is (x – 4) cm long. The second line is divided into three equal parts, each is (x + 2) cm long. a The lines are the same length. Write an equation to show this. b Solve the equation. c Work out the length of each line. 15 a Write down an expression for the length of the perimeter of the triangle. b Write down an expression for the length of the perimeter of the square. c The perimeters of the shapes are the same length. Write down an equation to show this. d Solve the equation. e Work out the lengths of the sides of each shape. x – 4 x + 2 t t + 6 t + 6 t + 6 t t t © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 400 2867_P001_680_Book_MNPB.indb 400 24/03/2020 15:33 4/03/2020 15:33